Dynamic Calibration Based on the Black-Scholes Option Pricing Model by Bayesian Method

Abstract

To improve the shortcomings of the classic Black-Scholes model, mainly on the constant volatility and normal distribution assumptions, this paper investigates the dynamic calibration method, which makes the expected return rate, volatility and interest rate become data-driven and time dependent. Based on the dynamic procedure, four distinct calibration models are proposed by using Bayesian method, among which Model I and Model II are used for comparison, Model III simultaneously uses the data of underlying asset, put and call options by introducing the bivariate normal distribution, and Model IV simplifies Model III by employing the put-call parity. The results of numerical experiments and empirical analysis illustrate that Model III is the most accurate but time consuming, while Model IV is the most efficient. Dynamic calibration method is also verified to be much more accurate in data fitting and option pricing than the commonly used global calibration. Overall, the dynamically calibrated Black-Scholes model can be regarded as an improvement on the classic Black-Scholes model, where model coefficients are functions of time. As a result, leptokurtic and negative skew distribution of log returns is regenerated, which makes the model more consistent with the data observed in real markets without an increase of the complexity.

Keywords

• Bayesian method
• dynamic calibration
• Black-Scholes model
• Markov Chain Monte Carlo
• volatility surface
• leptokurtosis